Quantum computing devices with Majorana Hexon qubits

ABSTRACT

Various embodiments of a modular unit for a topologic qubit and of scalable quantum computing architectures using such modular units are disclosed herein. For example, one example embodiment is a modular unit for a topological obit comprising  6  Majorana zero modes (MZMs) on a mesoscopic superconducting island. These units can provide the computational MZMs with protection from quasiparticle poisoning. Several possible realizations of these modular units are described herein. Also disclosed herein are example designs for scalable quantum computing, architectures comprising the modular units together with gates and reference arms (e.g., quantum dots, Majorana wires, etc.) configured to enable joint parity measurements to be performed for various combinations of two or four MZMs associated with one or two modular units, as well as other operations on the states of MZMs.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a divisional of U.S. Nonprovisional application Ser.No. 15/636,376, entitled “QUANTUM COMPUTING DEVICES WITH MAJORANA HEXONQUBITS” filed on Jun. 28, 2017, which claims the benefit of U.S.Provisional Application No. 62/376,386, entitled “MEASURING ANDMANIPULATING MAJORANA QUASIPARTICLE STATES USING THE STARK EFFECT” filedon Aug. 17, 2016, and U.S. Provisional Application No. 62/378,218,entitled “MEASURING AND MANIPULATING STATES OF NON-ABELIANQUASIPARTICLES VIA QUANTUM DOT HYBRIDIZATION ENERGY SHIFTS” filed onAug. 23, 2016; and U.S. Provisional Application No. 62/382,253, entitled“QUANTUM COMPUTING DEVICES WITH MAJORANA HEXON QUBITS” filed on Aug. 31,2016, all of which are hereby incorporated herein by reference in theirentirety.

This application is a divisional of U.S. Nonprovisional application Ser.No. 15/636,376, entitled “QUANTUM COMPUTING DEVICES WITH MAJORANA HEXONQUBITS” filed on Jun. 28, 2017, which is also a continuation in part ofU.S. Nonprovisional application Ser. No. 15/634,983, entitled “MEASURINGAND MANIPULATING STATES OF NON-ABELIAN QUASIPARTICLES VIA QUANTUM DOTHYBRIDIZATION ENERGY SHIFTS” filed on Jun. 27, 2017, which is herebyincorporated herein by reference in its entirety.

FIELD

This application relates generally to quantum computers. Morespecifically, the application concerns topologically protected quantumcircuits.

SUMMARY

Various embodiments of a modular unit for a topologic qubit and ofscalable quantum computing architectures using such modular units aredisclosed herein. For example, one example embodiment is a modular unitfor a topological quint comprising six Majorana zero modes (MZMs) on amesoscopic superconducting island. These units can provide thecomputational MZMs with protection from quasiparticle poisoning. Severalpossible realizations of these modular units are described herein. Alsodisclosed herein are example designs for scalable quantum computingarchitectures comprising the modular units together with gates andreference arms (e.g., quantum dots, Majorana wires, etc.) configured toenable joint parity measurements to be performed for variouscombinations of two or four MZMs associated with one or two modularunits, as well as other operations on the states of MZMs.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic block diagram of an example two-sided MajoranaHexon qubit.

FIG. 2 is a schematic block diagram of an example two-sided MajoranaHexon cubit with a reference arm.

FIG. 3 is a schematic block diagram of an example network of two-sidedMajorana Hexon qubits.

FIG. 4 is a schematic block diagram of an example one-sided MajoranaHexon qubit.

FIG. 5 is a schematic block diagram of an example network of one-sidedMajorana Hexon qubits.

FIG. 6 is a schematic block diagram showing symmetry of an exampleone-sided Majorana Hexon qubit layout.

FIG. 7 is a schematic block diagram showing an example network of linearMajorana Hexon qubits.

FIGS. 8-16 are diagrams illustrating further example Hexonconfigurations.

DETAILED DESCRIPTION I. General Considerations

Disclosed herein are representative embodiments of methods, apparatus,and systems for topological quantum devices, and in particular fortopologic qubits as can be used in a topological quantum computer. Thedisclosed methods, apparatus, and systems should not be construed aslimiting in any way. Instead present disclosure is directed toward allnovel and nonobvious features and aspects of the various disclosedembodiments, alone or in various combinations and subcombinations withone another. Furthermore, any features or aspects of the disclosedembodiments can be used alone or in various combinations andsubcombinations with one another. For example, one or more method actsfrom one embodiment can be used with one or more method acts fromanother embodiment and vice versa. The disclosed methods, apparatus, andsystems are not limited to any specific aspect or feature or combinationthereof, nor do the disclosed embodiments require that any one or morespecific advantages be present or problems be solved.

Although the operations of some of the disclosed methods are describedin a particular, sequential order for convenient presentation, it shouldbe understood that this manner of description encompasses rearrangement,unless a particular ordering is required by specific language set forthbelow. For example, operations described sequentially may in some casesbe rearranged or performed concurrently. Moreover, for the sake ofsimplicity, the attached figures may not show the various ways in whichthe disclosed methods can be used in conjunction with other methods.

Various alternatives to the examples described herein are possible. Forexample, some of the methods described herein can be altered by changingthe ordering of the method acts described, by splitting, repeating, oromitting certain method acts, etc. The various aspects of the disclosedtechnology can be used in combination or separately. Differentembodiments use one or more of the described innovations. Some of theinnovations described herein address one or more of the problems notedherein. Typically, a given technique/tool does not solve all suchproblems.

As used in this application and in the claims, the singular forms “a,”“an,” and “the” include the plural forms unless the context clearlydictates otherwise. Additionally, the term “includes” means “comprises.”Further, as used herein, the term “and/or” means any one item orcombination of any items in the phrase.

II. Overview of Disclosed Technology

Various embodiments of a modular unit for a topologic qubit and ofscalable quantum computing architectures using such modular units aredisclosed herein. For example, one example embodiment is a modular unitfor a topological qubit comprising six Majorana zero modes (MZMs) on amesoscopic superconducting island. Such a modular unit is sometimesreferred to herein as “Majorana Hexon qubit.” These units can providethe computational MZMs with protection from quasiparticle poisoning.Several possible realizations of these modular units are describedherein. Also disclosed herein are example designs for scalable quantumcomputing architectures comprising the modular units together with gatesand reference arms (e.g., quantum dots, Majorana wires, etc.) configuredto enable joint parity measurements to be performed for variouscombinations of two or four MZMs associated with one or two modularunits, as well as other operations on the states of MZMs. These can beconfigured in such a way to allow the generation of all the Cliffordgates with topological protection and non-Clifford gates (e.g., phasegate) without topological protection, thereby producing acomputationally universal gate set. Several possible realizations ofthese architectures are disclosed.

While the figures depict realizations of MZMs at the endpoints ofnanowires, the principles of the disclosed technology can also apply tomore general realizations of MZMs (e.g., 2DEGs) and are not specificallylimited to nanowire realizations.

Further, while the example embodiments discussed below focus on six MZMsper superconducting island, larger numbers of MZMs per island are alsopossible and considered to be within the scope of the disclosedtechnology.

Certain example embodiments of the disclosed modular units lead toplanar, two-dimensional, connected networks of qubits. Further, certainembodiments of the disclosed technology provide great flexibility inHexon manipulations (via braiding, of Majoranas) through the presence ofan extra ancilia. In further embodiments, the ancillas in the networkare integrated (and shared between one or more Hexons) in such a mannerthat additional ancillas are not necessary. For instance, in someembodiments, each Hexon can be manipulated or entangled with otherHexons without the need of additional qubits that would act as ancillas.

Further examples and details concerning the disclosed technology, aswell as other architectures with which the disclosed technology can beused, are described in T. Karzig et al., “Scalable Designs forQuasiparticle-Poisoning-Protected Topological Quantum Computation withMajorana Zero Modes,” arXiv: 1610.05289 (March 2017) and T. Karzig et al“Scalable Designs for Quasiparticle-Poisoning-Protected TopologicalQuantum Computation with Majorana Zero Modes,” Phys. Rev. B 95, 235305(2017), both of which are hereby incorporated herein by reference.

These and other embodiments will be explained in more detail in thedescription below.

III. Example Majorana Hexon Qubits

One example embodiment disclosed herein is a modular unit comprising sixor more MZMs situated on a single mesoscopic superconducting island.When the charging energy E_(c) of the island is large, it provides thecontained MZMs protection from quasiparticle poisoning.

Systems with MZMs are often referred to as “topological” or“topologically protected,” hut, more precisely, they are “symmetryprotected topological,” as detailed in Phys. Rev. B 87, 195451 (2013)[arXiv: 1212.6395]. In particular, their topological protection is basedon fermion parity symmetry, meaning the system's fermion parity must bepreserved in order to manifest topological protection of the statesencoded nonlocally in the MZMs. Consequently, MZM systems arepotentially vulnerable to stray electrons entering MZMs from outside thesystem—a quasiparticle poisoning. Such events constitute errors forquantum information encoded in the (nonlocal) fermion parities of MZMs.In particular, they take qubits out of the computational subspace, sincethey flip the fermion parities associated with the MZMs. Hence, it isdesirable to protect MZM systems used for quantum computation fromquasiparticle poisoning.

Six is the smallest number of MZMs that supports the combination of onecomputational qubit (which is encoded in four of the MZMs) and oneancillary pair of MZMs. This combination is particularly useful becausethe presence of an ancillary pair makes it possible to generate braidingtransformations using the “measurement-only” protocols detailed in U.S.Pat. No. 8,209,279: “Measurement-Only Topological Quantum Computation”and papers Phys. Rev. Lett. 101, 010501 (2008) [arXiv: 0802.0279] andAnnals Phys. 324, 787-826 (2009) [arXiv: 0808.1933] or the tunablecouplings methods, details of which can be found in e.g. Phys. Rev. B87, 035113 (2013) [arXiv: 1210.7929] and Phys. Rev. B 88, 035121 (2013)[arXiv: 1303.4379]. In other words, one can perform sequences oftopological charge (joint parity)measurements or sequences of tuningcouplings, rather than physically transporting the MZMs in order togenerate the braiding transformations on the qubit states encoded inMZMs. Thus, using the modular unit (a Majorana Hexon qubit) togetherwith the ability to perform measurements of certain pairs of MZMs and/orthe ability to control the tuning of couplings between the pairs ofMZMs, one can generate the single-qubit Clifford gates with topologicalprotection, while simultaneously protecting the qubit from quasiparticlepoisoning errors.

The joint fermion parity measurements of MZMs play be implemented, forexample, by conductance measurements, as described in Phys. Rev. Lett.116, 050501 (2016) [arXiv: 1509.05345], or quantum dot spectroscopy, asdescribed in U.S. Provisional Patent Application 62/378,218 and U.S.Provisional Patent Application 62/176,386. In this disclosure, the focusis primarily on measurement-only implementations of the braidingtransformations, but it should be recognized that the braidingtransformations can equivalently be performed by adiabatic evolution viachanging Majorana couplings.

Spanning the full set of single-qubit Clifford gates on a given MajoranaHexon qubit (or the full set of braiding transformations for thenon-ancillary MZMs on an island) involves a minimal set of pairs of MZMsupon which one can perform joint parity measurements. An efficientimplementation follows from allowing measurements of each pair of MZMswithin an island, though a subset of the pairs is sufficient. Practicalconstraints may limit or constrain which pairs can be measured.

Moreover, these modular units (Majorana Hexon qubits) are alsocompatible with generating entangling gates, e.g., two-qubit Cliffordgates, while simultaneously protecting the qubits from quasiparticlepoisoning, through the use of joint parity measurements of 4 MZMs (2MZMs from one Majorana Hexon qubit unit and 2 from another). Togetherwith the single qubit Clifford gates, these generate all (n-qubit)Clifford gates. More details on how such measurements generate theClifford gates can be found in the “Appendix—Hexon Details” sectionbelow. It should be noted that, as long as one is able to perform theappropriate measurements of MZMs, one ancillary pair of MZMs on anisland is sufficient for implementing entangling gates between twoqubits on separate islands (between two Majorana Hexon qubits), withoutthe need of extra ancillary MZMs, as detailed in the “Appendix—HexonDetails” section below.

It is known that the Clifford gates by themselves do not form acomputationally universal gate set, but that supplementing the Cliffordgates with a non-Clifford 1-qubit gate, e.g. the Π/8-phase gate yields acomputationally universal gate set. Such a gate can be produced from“magic states” by using measurements. Magic states can be generated in anumber of ways for Majorana systems. The apparatuses used to performmeasurements and/or tune couplings of MZMs mentioned in this disclosurecan be used to generate magic states. (These magic states will not betopologically protected, so they will likely require someerror-correction, e.g. by magic state distillation methods of Phys. Rev.A 71, 022316 (2005) [quant-ph/0403025]; if desired, one can also utilizecancelation schemes, such as those detailed in Phys. Rev. X 6, 31019(2016)[arXiv:1511.05161] to improve the fidelity of magic stategeneration, before distillation.) Example methods of generating magicstates with the described apparatus include the partial interferometrymethods detailed in U.S. Pat. No. 9,256,834: “Quantum Computers HavingPartial Interferometric Quantum Gates” and the following discussion ofperforming measurements of non-Pauli operators.

The next sections described in more detail several possiblearchitectures for realizing Majorana Hexon qubits in accordance with thedisclosed technology.

IV. Two-Sided Majorana Hexon Qubit

In accordance with one example embodiment, the two-sided Majorana Hexonqubit comprises three one-dimensional topological superconductors(1DTSs) (e.g. half-shell Al-coated InAs nanowires) that are joined by asuperconducting “backbone.” For purposes of this discussion and withoutlimitation, the horizontal direction is defined to be along the lengthof the 1DTSs and the vertical direction to be along the length of thebackbone (perpendicular to the 1DTSs). The backbone effectively joinsthe three 1DTSs to form a single island. (The directional orientationsused herein are for explanatory purposes, as the disclosed embodimentscan be rotated into different orientations.)

FIG. 1 is a schematic block diagram 100 illustrating this configuration.In the figures, and in addition to the color coding, reference numbersor symbols are provided to designate the various components inaccordance with the key at the bottom of each figure (e.g., s-wavesuperconductors 112, half-shell topological wires (or p-wavesuperconductors) 110, and Majorana zero mode (labelled some cases forease of illustration, and usually with respect to the half-shelltopological wire for P-wave superconductor) on which the MZMs reside,only a few examples of the components are highlighted by referencenumbers even though additional instances of the component areillustrated in the figure. One skilled in the art will recognize thoseadditional instances based on the example reference numbers illustrated.

The separate charging energy of the 1DTSs is exponentially suppressed bythe number of channels that connect the 1DTSs to the backbone. Thetypical number of channels in the superconductor of half-shell nanowirerealizations is expected to be larger than 1000.

The MZMs reside at the ends of the 1DTSs of length L. The remaininghybridization splitting between MZMs will be exponentially suppressed byexp(−L/ξ), where ξ is the effective coherence length in the 1DTSs. Notethat for a short backbone, L not only controls the hybridization withina 1DTS, but also between MZMs of different 1DTSs (for a sufficientlyinsulating substrate).

A two-sided Majorana Hexon qubit may comprise 1DTSs that are closelyspaced in the vertical direction, but long in the horizontal direction(to minimize unwanted hybridization). (Again, the directionalorientations used herein are for explanatory purposes, as the disclosedembodiments can be rotated into different orientations.) Close verticalspacing allows easy access for measurements on any two of the threenearby MZMs on a given side (left or right) of the two-sided MajoranaHexon qubit. In particular, and in some particular embodiments, ameasurement scheme is employed where each of the ends of the 1DTSs canbe connected to (and disconnected from) a semiconducting wire that isaligned perpendicular to the 1DTSs, as shown in schematic block diagram300 of FIG. 3. The connection can be tuned by depletion gates. Anadditional set of gates can split the semiconducting, wire into separatequantum dots with full control over the coupling strength between thesequantum dots. This setup therefore allows the connection of any two MZMsto coupled quantum dots. The measurement then proceeds by using thequantum clots as input and output for conductance measurements ordirectly probing the energy levels of the quantum dots (in the lattercase also using a single dot suffices).

As mentioned above, measuring pairs within the groups of three nearbyMZMs (on the left or right of the unit) is desirably complemented bymeasurements of pairs of MZMs selected from the opposite sides of theunit, which are separated by a distance of the order L. Theselong-distance measurements can be implemented by adding a coherentreference arm of length L (for example in the form of an additional,isolated 1DTS), as shown in schematic block diagram 200 in FIG. 2. Whenthe reference arm is coupled to the vertical semiconducting wire bysimilar controls as the three connected 1DTSs, the flexibility ofselectively coupling any of the nearby MZMs to the verticalsemiconducting wire allows for performing all long-distancemeasurements. If the distance of the MZMs to the reference arm becomesan issue for the visibility of the measurement, it is also possible toonly perform long-distance measurements involving MZMs that are at mosttwice the distance between neighboring 1DTSs apart from the referencearm. Note that another way of performing the long-distance measurementsbecomes possible if the backbone can be selectively weakened to (atleast) partially disconnect one of the 1DTSs from the other two. In thatcase, the separate charging energy of the disconnected 1DTS splits thetwo fermionic parity states of the MZMs on the corresponding 1DTS, whichallows for a measurement via charge sensing.

Entangling gates can be implemented by operations that includemeasurements of the joint parity of 4 MZMs, two from each qubit (ondifferent islands). Again, these measurements are possible viaconductance or spectroscopic measurements. Such 4 MZM measurementsinvolving two horizontally adjacent two-sided Majorana Hexon qubits(denoted by left and right) can be performed by using the same verticalsemiconducting wire at the right end of the left two-sided MajoranaHexon qubit and the left end of the right two-sided Majorana Hexonqubit, as seen in FIG. 3. The two dots used in the measurement wouldboth be defined within the vertical semiconducting wire, and would bedisconnected except for one hybridization path through MZMs of the leftHexon qubit and another through MZMs of the right Hexon qubit. A 4 MZMmeasurement involving two vertically adjacent two-sided Majorana Hexonqubits (denoted upper and lower) can be performed by making the verticalsemiconducting wire extended such that the upper and lower two-sidedMajorana Hexon qubits share an uninterrupted part of the semiconductingwire. One of the dots used in the measurement would then be defined onthe left shared vertical semiconducting wire, while the other is on theright shared vertical semiconducting wire. For these measurements, extrareference arms would be disconnected from the two-sided Majorana Hexonqubits so that the two measured qubits can only hybridize via the upperor lower MZM states.

V. One-Sided Majorana Hexon Qubit

An example one-sided Majorana Hexon qubit comprises 6 1DTSs (of lengthL) that are connected by a superconducting backbone at one side(backbone side). FIG. 4 is a schematic block diagram 400 illustratingthis example one-sided Majorana Hexon qubit. The short vertical distancebetween neighboring 1DTS can lead to a strong hybridization of the 6MZMs located at the backbone side of the 1DTSs, effectively leaving only6 MZMs at the other side (MZM side). One advantage of this design overthe two-sided Majorana Hexon qubit is that it provides topologicalprotection of the MZMs corresponding to length 2L, rather than L, for1DTSs of length L. Since the capacitance of wires (with length muchlarger than the diameter) grows linearly with length, but onlylogarithmically with diameter, it is estimated that in the elongateddesign of Majorana Hexon qubits, the capacitance will be decreased whenthe islands are made half as long, while doubling the width. Theone-sided Majorana Hexon qubit is therefore a way to increase thecharging energy of the 6-Majorana islands (which increases theprotection from quasiparticle poisoning).

Aside from possible changes in the overall charging energy of eachisland, operations for one-sided Majorana Hexon qubits can beimplemented in a very similar manner as for the two-sided Majorana Hexonqubits. The Majorana side of the island is again connected to a verticalsemiconducting wire that can be gated into quantum dots with tunableconnections of the resulting quantum dots to the 6 MZMs. Since all 6MZMs are now located at the same side, there are no long distances ofthe order of L between any pair of MZMs. This allows one to measure thejoint fermionic parity of any pair of the 6 MZMs (similar to thesituation for the three MZMs at each side of the two-sided MajoranaHexon qubit), without the need of a long (length of order L) referencearm.

An example network of one-sided Majorana Hexon qubits can comprise pairsof one-sided Majorana Hexon qubits (one-sided pairs) aligned at thebackbone side, with the Majorana side of the left one-sided MajoranaHexon qubit being on the left and the Majorana side of the rightone-sided Majorana Hexon Qubit being on the right. In one particularembodiment, one reference arm of length 2L is located above and below aone-sided pair, as shown in schematic block diagram 500 of FIG. 5. Thisstructure can then be repeated in the vertical direction, forming avertical column in the array. The adjacent vertical columns can bevertically displaced by 4 1DTSs, as shown in FIG. 5. This way, the rightpart of the one-sided pair can connect (via the upper two MZMs) to thelower two MZMs of a one-sided. Majorana Hexon qubit in the adjacentvertical column. Similarly, the lower two MZMs can connect to the uppertwo MZMs of a one-sided Majorana Hexon qubit in the adjacent verticalcolumn. These connections allow operations (e.g., joint paritymeasurements of 4 MZMs) that entangle the right part of each column withthe left part of the next column. In order to also allow connectionsbetween the right and left part of each vertical column, the abovementioned extra reference arms can be used. The latter connect the rightpart of a one-sided pair with the left part of the one-sided pair of thenext-to-nearest column (at the same horizontal position). The right andthe left parts of the jth column are now connected to each other throughtheir respective connections to the left part of the j+1th column andthe right part of the j−1th column, which are in turn connected to eachother (see FIG. 5). Thus, a fully connected array can be constructed, asshown in FIG. 5 and in schematic block diagram 600 in FIG. 6.

VI. Linear Majorana Hexon Qubit

An example linear Majorana Hexon qubit comprises one long 1DTS (lengthL) that is partitioned into three topological superconducting regionsseparated by two normal non-topological (s-wave) superconductingregions. FIG. 7 is a schematic block diagram 700 illustrating an exampleof a network of linear Majorana Hexons. The non-topologicalsuperconducting regions can be created from a long 1DTS by gating. Sincegating does not affect the superconducting part, the superconducting“backbone” is naturally provided by the normal regions of the IDTS thatconnect the three topological superconducting regions, effectivelyforming a one-dimensional island. (In a realization of half-shell wires,the superconducting backbone would be the superconducting shell that ispresent throughout the entire half shell wire of length L.) In order forthe system to realize MZMs at the end of the topological superconductingregions, the Lengths L_(top) of the topological superconducting regionsas well as the lengths L_(normal) of the normal superconducting regionsare desirably sufficiently long as compared with the respectivecoherence lengths of these regions. e.g. L_(top)>>ξ_(top) andL_(normal)>>ξ_(normal). Assuming ξ_(top)≈ξ_(normal) andL_(region)=L_(top)≈L_(normal), so that the overall length of theone-dimensional island is L=5L_(region), this configuration providestopological protection of the MZMs corresponding to L_(region)=L/5. Fromsimilar arguments as in the previous section, the charging energy oflinear Majorana Hexon qubits is smaller than both the two-sided andone-sided versions of Majorana Hexon qubits. A possible advantage of thelinear Majorana Hexon qubit design is the absence of an additionalstructure to provide the superconducting backbone.

Due to the linear layout of this design, there are no MZM pairs thatseparated by at least L_(top) or L_(normal). In this sense, measurementscannot be performed by coupling nearby MZMs, as was used in the previousdesigns. Instead, all measurements require reference arms of sufficientlength. One example layout, shown in FIG. 7, uses reference arms oflength L_(top)+L_(normal). The configuration of reference arms shown inFIG. 7 allows one to measure the joint parity of the following pairs ofthe MZMs on a given Hexon unit, labeled 1 to 6 from left to right: (1,2), (1, 3), (2, 3), (3, 4), (3, 5), and (4, 5). Additionally, theconfiguration of reference arms allows one to measure the joint parityof the group of 4 MZMs composed of: (L5, L6, R1, R2) for twohorizontally adjacent linear Majorana Hexon qubits (L and R indicatingthe left and right unit, respectively); and either (U5, U6, D5, D6) or(U3, U4, D3, D4) for two vertically adjacent linear Majorana. Hexonqubits (U and D indicating the upper and lower unit, respectively),depending on which rows of the array the qubits are in. These operationsare sufficient to perform all Clifford operations, as outlined in the“Appendix—Hexon Details” section below.

VII. Quantum Computing Device Design Considerations

In the sections above, several designs for assembling the differenttypes of Majorana Hexon qubits into scalable quantum computing deviceswere disclosed. In general, these designs comprise periodic arrays ofthe superconducting islands hosting 6 MZMs, together with quantum dotsand reference arms that can be used to perform the desired measurementsand operations on states of MZMs. The presented designs all havereasonable fabrication requirements, but they lie in different areas ofdesign “parameter space,” so the different designs may afford differentadvantages, e.g. in terms of fabrication and/or device characteristicsand operation.

One parameter to consider is the charging energy E_(c) ofsuperconducting island. Charging energy is a geometric quantity that isroughly inversely proportional to the length scale L of the object, e.g.E_(c)˜1/L. It also is affected by the local electrical environment.E_(c) is desirably selected to balance two opposing criteria: it shouldbe large enough to suppress quasiparticle poisoning, which may take theform of stray electrons entering the system through a MZM; and it shouldbe small enough that read-out signals of the measurements essential tothe desired manipulation of the system are strong, enough. Per U.S.Provisional Application No. 62/378,218, the quantum dot energy shiftsplaying a role in measurements are proportional to 1/E_(c). Othermeasurement protocols show a scaling 1/(E_(c))². In either case E_(c)cannot be too large or the required signal will not be detectable in ashort period of time (e.g. computer clock cycle). The designs disclosedherein have length scales ranging over a factor of approximately 5. Thedesign “Linear Majorana Hexon Qubit” has the smallest, next is the“2-Sided Majorana Hexon Qubit,” and largest is the “One-Sided MajoranaHexon Qubit.” Other design parameters include simplicity of fabrication.In particular, the Linear Majorana Hexon Qubit has the simplicity of notrequiring a strong superconducting weld (large E_(J)) joiningconstituent pieces. All designs can realize the Clifford gate set in anefficient manner via measurement-only methods.

The designs disclosed herein can be implemented by using well-developedbuilding blocks in this area of micro-technology. For example, incertain embodiments, the designs can be realized by building a structureof p-wave superconducting nanowires (Majorana nanowires) on aninsulating wafer and connecting MZMs hosted by these wires via quantumdots built from gated semiconductor overlying the wires and portions ofthe wafer. Wires of the required p-wave type were described in NatureNanotechnology 10, 232 (2015) [arXiv: 1411.6255] and Nature Materials14, 400 (2015) [arXiv: 1411.6254]. The example designs use MZMs that canbe localized either at the ends of wires or in middle regions, wherethey may be created by gating an internal segment of the wire so it istuned out of the topological regime, e.g. as described in Nature Physics7, 412 (2011) [arXiv: 1006.4395]. However, the general principles of thedisclosed technology and the particular designs illustrated are notlimited to this style of fabrication. For example, it is also possibleto fabricate our layouts by gating two dimensional electron gasses(2DEGs) and proximitizing them with a superconductor, as described inPhys. Rev. B 93, 155402 (2016) [arXiv: 1511.01127]. All fabricationmethods of these designs are considered to be within the scope of thedisclosed technology.

VIII. Further Representative Embodiments

This section describes various example representative embodiments of thedisclosed technology. These embodiments should not be construed aslimiting, as they can be modified in arrangement and detail withoutdeparting from the principles of the disclosed technology.

The disclosed embodiment include embodiments of asymmetry-protected-topological and quasiparticle-poisoning-protectedMajorana Hexon qubit.

For instance, one example embodiment as described herein is a two-sidedMajorana Hexon qubit, comprising: three topological superconductingnanowires, each of the topological superconducting nanowires having arespective first end at which a respective first Majorana zero moderesides and a respective second end, opposite the first respective end,where a respective second Majorana zero mode resides; and asuperconducting backbone element connected to the three topologicalsuperconducting nanowires, the superconducting backbone element beinglocated between the respective first ends and the respective second endsof the three topological superconductive nanowires. In someimplementations, the three topological superconducting nanowires arelocated on a superconducting island having a charging energy sufficientto prevent quasiparticle poisoning.

In some examples, the superconducting backbone element is orientedtransverse to the three topological superconducting nanowires. Infurther examples, the three topological superconducting nanowires arehalf-shell topological wires or p-wave superconductors, and wherein thesuperconducting backbone element is an s-wave superconductor.

In some examples, the two-sided Majorana Hexon qubit further comprises afourth topological superconducting nanowire selectively coupled to therespective first ends of the three topological superconducting nanowiresand having a second end selectively coupled to the respective secondends of the three topological superconducting nanowires. In suchembodiments, the first end of the fourth topological superconductingnanowire can be selectively coupled to the respective first ends of thethree topological superconducting nanowires via a first-endsemiconductive wire, and the second end of the fourth topologicalsuperconducting nanowire can be selectively coupled to the respectivesecond ends of the three topological superconducting nanowires via asecond-end semiconductive wire.

In further examples, the two-sided Majorana Hexon qubit of can furthercomprise a set of first-end depletion gates arranged to provideselective quantum-dot couplings between the fourth topologicalsuperconducting nanowire and any one or more of the first ends of thethree topological superconducting nanowires via the first-endsemiconductive wire; and a set of second-end depletion gates arranged toprovide selective quantum-dot couplings between the fourth topologicalsuperconducting nanowire and any one or more of the second ends of thethree topological superconducting nanowires via the second-endsemiconductive wire.

Further embodiments comprise a network comprising multiple instances ofany of the two-sided Majorana Hexon qubits disclosed. In suchembodiments, at least one of the first-end semiconductive wire or thesecond-end semiconductive wire can be shared with one or moreneighboring instances of the two-sided Majorana Hexon qubit. Forinstance, the one or more neighboring instances of the two-sidedMajorana Hexon qubit can be neighboring, along a horizontal direction, avertical direction, or both.

Another example embodiment as described herein is a one-sided MajoranaHexon qubit, comprising: six topological superconducting nanowires, eachof the topological superconducting nanowires having a respectiveMajorana-zero-mode (MZM) end at which a respective first MZM resides anda respective backbone end opposite the MZM end; and a superconductingbackbone element connected to the six topological superconductingnanowires, the superconducting backbone element being connected to thesix topological superconducting nanowire at the respective backbone endsof the six topological superconductive nanowires.

In some examples, the superconducting backbone element is orientedtransverse to the six topological superconducting nanowires. In furtherexamples, the six topological superconducting nanowires are half-shelltopological wires or p-wave superconductors, and the superconductingbackbone element is an s-wave superconductor. In certain examples, thesix topological superconducting nanowires are located on asuperconducting island having a charging energy sufficient to preventquasiparticle poisoning.

Further embodiments comprise a network comprising multiple instances ofany of the one-sided Majorana Hexon qubits as described above. In someembodiments, a first instance and a second instance of the one-sidedMajorana Hexon qubits are arranged such that their respectivesuperconducting backbone elements face one another and their respectivesix MZM ends are in opposite directions from one another. In suchembodiments, the network can further comprise a first reference-armtopological superconducting nanowire having a first end selectivelycoupled to each of the respective MZM ends of the first instance of theone-sided Majorana Hexon qubit and having a second end selectivelycoupled to each of the respective MZM ends of the second instance of theone-sided Majorana hexon qubit. Still further, some exampleimplementations further comprise a second reference-arm topologicalsuperconducting nanowire having a first end selectively coupled to eachof the respective MZM ends of the first instance of the one-sidedMajorana Hexon qubit and having a second end selectively coupled to eachof the respective MZM ends of the second instance of the one-sidedMajorana hexon qubit, the second reference-arm topologicalsuperconducting nanowire being located on an opposite side of the firstinstance and the second instance of the one-side Majorana Hexon qubitthan the first reference-arm topological superconducting nanowire.

In some examples, the first end of the first reference-arm topologicalsuperconducting nanowire is selectively coupled to each of therespective MZM ends of the stance of the one-sided Majorana Hexon qubitvia a first-end semiconductive wire, and the second end of the firstreference-arm topological superconducting nanowire is selectivelycoupled to each of the respective MZM ends of the second instance of theone-sided Majorana Hexon qubit via a second-end semiconductive wire. Insuch examples, the network can further comprise a set of first-enddepletion gates arranged to provide a selective quantum-dot couplingbetween the first end of the first reference-arm topologicalsuperconducting nanowire and any one or more of the MZM ends of thefirst instance of the one-sided Majorana Hexon qubit via the first-endsemiconductive wire; and a set of second-end depletion gates arranged toprovide a selective quantum-dot coupling between the second end of thefirst reference-arm topological superconducting nanowire and any one ormore of the MZM ends of the second instance of the one-sided MajoranaHexon qubit via the second-end semiconductive wire. In someimplementations, at least one of the first-end semiconductive ware orthe second-end semiconductive wire is shared with one or moreneighboring instances of the one-sided Majorana Hexon qubit. Forinstance, the one or more neighboring instances of the one-side MajoranaHexon qubit can be neighboring along a horizontal direction, a verticaldirection, or both.

Further example embodiments described herein include a linear MajoranaHexon qubit, comprising: a topological superconducting nanowirepartitioned into regions, including: three MZM topologicalsuperconducting regions, each having a respective first end at which arespective first Majorana zero mode resides and a respective second end,opposite the first respective end, where a respective second Majoranazero mode resides; and two s-wave superconducting regions that separatethe three MZM topological superconducting regions from one another.

In some examples, the s-wave superconducting regions are created bygating junctions between the MZM topological superconducting regions andthe s-wave superconducting regions. In further examples, the topologicalsuperconducting nanowire is located on a superconducting island having acharging energy sufficient to prevent quasiparticle poisoning.

Further embodiments comprise two or more instances of the linearMajorana Hexon qubit as described above. For instance, in some exampleimplementations, one or more neighboring instances of the linearMajorana hexon obit are selectively coupleed one another bysuperconductive reference arms having tunable depletion gates configuredto provide quantum dot couplings between the neighboring instances. Infurther examples, the one or more neighboring, instances of the one-sideMajorana Hexon qubit are neighboring, along a horizontal direction, avertical direction, or both.

IX. Appendix—Hexon Details

As explained herein, the full set of single-qubit Clifford gates can begenerated on the computational qubit encoded in a single hexon given anappropriate minimal set of joint parity measurements of pairs of MZMs.One can diagrammatically represent the topological state of a hexon asshown in FIG. 8.

More specifically, FIG. 8 is a diagrammatic representation 800 of thetopological states (degenerate ground states) of a hexon. The center twoMZMs γ₃ and γ₄ fuse to even fermion parity, forming the ancillary pairof MZMs. The left and right pairs of MZMs both fuse to a=0 or 1, whichcorrespond to even or odd fermion parity, respectively. These outerpairs of MZMs form the computational qubit. The fusion channel a labelsthe qubit basis state.

In FIG. 8, the MZMs γ_(j) with j=1, . . . , 6 are labeled from left toright. The diagram may be interpreted as follows: The center two MZMs γ₃and γ₄, forming the ancillary pair, fuse to even fermion parity (p₃₄=1).The left-most and the right-most pairs of MZMs, γ₁ and γ₂, and γ₅ andγ₆, respectively, forming the computational qubit, have the same fusionchannel a=0 (even fermion parity) or 1 (odd fermion parity). That thefusion channel a labels the qubit basis states|0

=|p ₁₂ =p ₅₆=−1

  (1)|1

=|p ₁₂ =p ₅₆=+1

  (2)The total fusion channel of the four MZMs forming the computationalqubit is even fermion parity (p₁₂p₅₆=1).

In this appendix, the following equation is proven:Π₀ ⁽³⁴⁾Π₀ ⁽³⁵⁾Π₀ ⁽⁵⁶⁷⁸⁾Π₀ ⁽⁴⁵⁾Π₀ ⁽³⁴⁾ ∝W ⁽⁵⁶⁷⁸⁾⊗Π₀ ⁽³⁴⁾,  (3)

and it is explained how to shuttle computational MZMs through the qubit.(In the following diagrammatic analysis, unimportant overall constantsare neglected.)

For the two-qubit entangling gate

$\begin{matrix}{{W = \begin{pmatrix}1 & 0 & 0 & 0 \\0 & i & 0 & 0 \\0 & 0 & i & 0 \\0 & 0 & 0 & 1\end{pmatrix}},} & (4)\end{matrix}$one begins by considering the qubit basis states of two hexons in theconfigurations as shown in diagram 900 of FIG. 9.

As in FIG. 8, b ε {0,1} label the fermion parity even or odd states ofthe outermost pairs MZMs in a given hexon.

Projecting the fusion channel of MZMs 4 and 5 to vacuum (e.g., usingforced measurement) results in diagram 1000 of FIG. 10, where the jagged(wiggly) line denotes fusion to a fermion.

Applying the four-MZM projector Π₀ ⁽⁵⁶⁷⁸⁾ to the above superpositionyields block diagram 1100 of FIG. 1100.

Then, projecting MZMs 3 and 5 to the vacuum channel gives diagram 1200of FIG. 1200.

This step utilizes the diagrammatic braiding relation of MZMs or Isinganyons as shown in diagram 1300 of FIG. 1300.

Finally, projecting MZMs 3 and 4 to the vacuum channel gives diagram1400 of FIG. 1400, which is the desired entangling gate.

An alternative derivation of the equation introduced in this appendixabove can be performed by explicitly multiplying the projectors writtenin terms of Majorana operators, as follows

                                            (5) $\begin{matrix}{{\Pi_{0}^{(34)}\Pi_{0}^{(35)}\Pi_{0}^{(5678)}\Pi_{0}^{(45)}\Pi_{0}^{(34)}} = {\Pi_{0}^{(34)}\begin{matrix}{1 - {i\;\gamma_{3}\gamma_{5}}} \\2\end{matrix}\begin{matrix}{1 - {i\;\gamma_{5}\gamma_{6}\gamma_{7}\gamma_{8}}} \\2\end{matrix}\begin{matrix}{1 - {i\;\gamma_{4}\gamma_{5}}} \\2\end{matrix}\Pi_{0}^{(34)}}} \\{= {2^{- 3}{\Pi_{0}^{(34)}\left( {1 - {i\;\gamma_{3}\gamma_{5}} - {i\;\gamma_{4}\gamma_{5}} + {\gamma_{3}\gamma_{4}} - {\gamma_{5}\gamma_{6}\gamma_{7}\gamma_{8}} +} \right.}}} \\{\left. {{i\;\gamma_{3}\gamma_{6}\gamma_{7}\gamma_{8}} - {i\;\gamma_{4}\gamma_{6}\gamma_{7}\gamma_{8}} + {\gamma_{3}\gamma_{4}\gamma_{5}\gamma_{6}\gamma_{7}\gamma_{8}}} \right)\Pi_{0}^{(34)}} \\{= {2^{- 3}\left( {1 + {\gamma_{3}\gamma_{4}} - {\gamma_{5}\gamma_{6}\gamma_{7}\gamma_{8}} + {\gamma_{3}\gamma_{4}\gamma_{5}\gamma_{6}\gamma_{7}\gamma_{8}}} \right)\Pi_{0}^{(34)}}} \\{= {2^{- 3}\left. \sqrt{}2 \right.\begin{pmatrix}{1 + i} \\\left. \sqrt{}2 \right.\end{pmatrix}\left( {1 + {i\;\gamma_{5}\gamma_{6}\gamma_{7}\gamma_{8}}} \right)\Pi_{0}^{(34)}}} \\{= {\begin{matrix}1 \\4\end{matrix}e^{i\;{\pi/4}}{W^{(5678)} \otimes {\Pi_{0}^{(34)}.}}}}\end{matrix}$

Here, Π₀ ⁽³⁴⁾i_(γ3γj)Π₀ ⁽³⁴⁾=Π₀ ⁽³⁴⁾i_(γ4γj)Π₀ ⁽³⁴⁾=0≠3 or 4 and Π₀ ⁽³⁴⁾projects i_(γ3γ4)=−1.

Finally, 1500 and 1600 of FIGS. 15 and 16 illustrate how thecomputational MZMs may be shuttled through the cubit using anyonicteleportation.

X. Concluding Remarks

Further examples and details concerning the disclosed technology, aswell as other architectures with which the disclosed technology can beused, are described in T. Karzig et al., “Scalable Designs forQuasiparticle-Poisoning-Protected Topological Quantum Computation withMajorana Zero Modes,” arXiv:1610.05289 (March 2017) and T. Karzig etal., “Scalable Designs for Quasiparticle-Poisoning-Protected TopologicalQuantum Computation with Majorana Zero Modes,” Phys. Rev. B 95, 235305(2017), both of which are hereby incorporated herein by reference.

Having described and illustrated the principles of the disclosedtechnology with reference to the illustrated embodiments, it will berecognized that the illustrated embodiments can be modified inarrangement and detail without departing from such principles. In viewof the many possible embodiments to which the principles of thedisclosed invention may be applied, it should be recognized that theillustrated embodiments are only preferred examples of the invention andshould not be taken as limiting the scope of the invention.

What is claimed is:
 1. A one-sided Majorana Hexon qubit, comprising: sixtopological superconducting nanowires, each of the topologicalsuperconducting nanowires having a respective Majorana-zero-mode (MZM)end at which a respective first MZM resides and a respective backboneend opposite the MZM end; and a superconducting backbone elementconnected to the six topological superconducting nanowires, thesuperconducting backbone element being connected to the six topologicalsuperconducting nanowires at the respective backbone ends of the sixtopological superconductive nanowires.
 2. The one-sided Majorana Hexonqubit of claim 1, wherein the superconducting backbone element isoriented transverse to the six topological superconducting nanowires. 3.The one-sided Majorana Hexon qubit of claim 1, wherein the sixtopological superconducting nanowires are half-shell topological wiresor p-wave superconductors, and wherein the superconducting backboneelement is an s-wave superconductor.
 4. The one-sided Majorana Hexonqubit of claim 1, wherein the six topological superconducting nanowiresare located on a superconducting island having a charging energysufficient to prevent quasiparticle poisoning.
 5. A network comprisingmultiple instances of the one-sided Majorana Hexon qubit of claim 1,wherein a first instance and a second instance of the one-sided MajoranaHexon qubits are arranged such that their respective superconductingbackbone elements face one another and their respective six MZM ends arein opposite directions from one another, and wherein the network furthercomprises a first reference-arm topological superconducting nanowirehaving a first end selectively coupled to each of the respective MZMends of the first instance of the one-sided Majorana Hexon qubit andhaving a second end selectively coupled to each of the respective MZMends of the second instance of the one-sided Majorana hexon qubit. 6.The network of claim 5, further comprising a second reference-armtopological superconducting nanowire having a first end selectivelycoupled to each of the respective MZM ends of the first instance of theone-sided Majorana Hexon qubit and having a second end selectivelycoupled to each of the respective MZM ends of the second instance of theone-sided Majorana hexon qubit, the second reference-arm topologicalsuperconducting nanowire being located on an opposite side of the firstinstance and the second instance of the one-side Majorana Hexon qubitthan the first reference-arm topological superconducting nanowire. 7.The network of claim 5, wherein the first end of the first reference-armtopological superconducting nanowire is selectively coupled to each ofthe respective MZM ends of the first instance of the one-sided MajoranaHexon qubit via a first-end semiconductive wire, and wherein the secondend of the first reference-arm topological superconducting nanowire isselectively coupled to each of the respective MZM ends of the secondinstance of the one-sided Majorana Hexon qubit via a second-endsemiconductive wire.
 8. The network of claim 7, further comprising: aset of first-end depletion gates arranged to provide a selectivequantum-dot coupling between the first end of the first reference-armtopological superconducting nanowire and any one or more of the MZM endsof the first instance of the one-sided Majorana Hexon qubit via thefirst-end semiconductive wire; and a set of second-end depletion gatesarranged to provide a selective quantum-dot coupling between the secondend of the first reference-arm topological superconducting nanowire andany one or more of the MZM ends of the second instance of the one-sidedMajorana Hexon qubit via the second-end semiconductive wire.
 9. Thenetwork of claim 8, wherein at least one of the first-end semiconductivewire or the second-end semiconductive wire is shared with one or moreneighboring instances of the one-sided Majorana Hexon qubit, wherein theone or more neighboring instances of the one-side Majorana Hexon qubitare neighboring along a horizontal direction, a vertical direction, orboth.